On Some Quasilinear Elliptic Systems with Singular and Sign-Changing Potentials

Using variational methods, we study the existence and nonexistence of nontrivial weak solutions for the quasilinear elliptic system $$\left\{\begin{array}{ll}- {\rm div}(h_1(|\nabla u|^2)\nabla u) = \frac{\mu}{|x|^2}u + \lambda F_u(x, u, \upsilon)\quad {\rm in}\,\Omega,\\- {\rm div}(h_2(|\nabla \upsilon|^2)\nabla \upsilon) = \frac{\mu}{|x|^2}\upsilon + \lambda F_\upsilon(x,u,\upsilon)\quad {\rm in}\,\Omega,\\u = \upsilon = 0 \qquad \qquad \qquad \qquad \qquad \qquad {\rm in}\, \partial\Omega, \end{array}\right.$$ where \({\Omega \subset \mathbb{R}^N,N \geq 3}\) , is a bounded domain containing the origin with smooth boundary \({\partial \Omega ; h_i, i = 1, 2}\) , are nonhomogeneous potentials; \({(F_u, F_v) = \nabla F}\) stands for the gradient of a sign-changing C ¹-function \({F : \Omega \times \mathbb{R}^2 \to \mathbb{R}}\) in the variable \({{w = (u, v) \in \mathbb{R}^2}}\) ; and λ and μ are parameters.     

Sign-definiteness of q-eigenfunctions for a super-linear p-Laplacian eigenvalue problem

We consider the following q-eigenvalue problem for the p-Laplacian $$\left\{\begin{array}{ll}-{\rm div}\big( |\nabla u|^{p-2}\nabla u\big) = \lambda \|u\|_{L^{q}(\Omega)}^{p-q}|u|^{q-2}u \quad \quad\, {\rm in} \,\,\,\, \Omega\\ \quad\quad\quad \quad \quad \quad u = 0 \quad\qquad\qquad \quad\quad \,\,{\rm on } \,\,\,\, \partial\Omega,\end{array}\right.$$ where \({\lambda\in\mathbb{R},}\) p > 1, Ω is a bounded and smooth domain of \({\mathbb{R}^{N},}\) N > 1, \({1\leq q < p^{\star}}\) , \({p^{\star}=\frac{Np}{N-p}}\) if p < N and \({p^{\star}=\infty}\) if \({p\geq N.}\) Let λ _q denote the first q-eigenvalue. We prove that in the super-linear case, \({p < q < p^{\star},}\) there exists \({\epsilon_{q}>0}\) such that if \({\lambda\in(\lambda_{q},\lambda _{q}+\epsilon_{q})}\) is a q-eigenvalue, then any corresponding q-eigenfunction does not change sign in Ω. As a consequence of this result we obtain, in the super-linear case, the isolatedness of λ _q for those Ω such that the Lane–Emden problem $$\left\{\begin{array}{ll}-{\rm div}\big(|\nabla u|^{p-2}\nabla u\big) = |u|^{q-2}u \qquad\quad\quad\quad \,\,{\rm in}\,\,\,\Omega\\ \quad\quad\quad \quad \quad \quad u = 0 \quad\qquad\qquad \quad\quad \,{\rm on } \,\,\, \partial\Omega,\end{array}\right.$$ has exactly one positive solution.     

Existence and Multiplicity Results for Some Elliptic Systems in Unbounded Cylinders

We study the following nonlinear elliptic system of Lane–Emden type $$\left\{\begin{array}{ll} -\Delta u = {\rm sgn}(v) |v| ^{p-1} \qquad \qquad \qquad \; {\rm in} \; \Omega , \\ -\Delta v = - \lambda {\rm sgn} (u)|u| \frac{1}{p-1} + f(x, u)\; \; {\rm in}\; \Omega , \\ u = v = 0 \qquad \qquad \qquad \quad \quad \;\;\;\;\; {\rm on}\; \partial \Omega , \end{array}\right.$$ where ${\lambda \in \mathbb{R}}$ . If ${\lambda \geq 0}$ and ${\Omega}$ is an unbounded cylinder, i.e., ${\Omega = \tilde \Omega \times \mathbb{R}^{N-m} \subset \mathbb{R}^{N}}$ , ${N - m \geq 2, m \geq 1}$ , existence and multiplicity results are proved by means of the Principle of Symmetric Criticality and some compact imbeddings in partially spherically symmetric spaces. We are able to state existence and multiplicity results also if ${\lambda \in \mathbb{R}}$ and ${\Omega}$ is a bounded domain in ${\mathbb{R}^{N}, N \geq 3}$ . In particular, a good finite dimensional decomposition of the Banach space in which we work is given.     

Strong Regularizing Effect of a Gradient Term in the Heat Equation with a Weight

We deal with the following parabolic problem, $$(P)\left\{\begin{array}{lll} u_t - \Delta{u} + |\nabla{u}|^q \quad=\quad \lambda{g}(x)u + f(x, t),\quad u > 0 \; {\rm in} \; \Omega \; \times \; (0, T),\\ \qquad\quad\quad\; u(x, t) \quad=\quad 0 \quad{\rm on}\; {\partial}{\Omega}\; \times ; (0, T),\\ \qquad\quad\quad\; u(x, 0) \quad=\quad u_{0}(x), \quad x \in {\Omega},\end{array}\right.$$ where is a bounded regular domain or ${\Omega = \mathbb{R}^N}$ , ${1 < q \leq 2, \lambda > 0\; {\rm and}\; f \geq 0, u_{0} \geq 0}$ are in a suitable class of functions. We give assumptions on g with respect to q for which for all λ >  0 and all ${f \in L^1(\Omega_T ), f \geq 0}$ , problem (P) has a positive solution. Under some additional conditions on the data, the Cauchy problem and the asymptotic behavior of the solution are also considered.     

Regularity lifting of weak solutions for nonlinear sub-Laplace equations on homogeneous groups

Let G be a homogeneous group, and let X ₁, X ₂, … , X _m be left invariant real vector fields being homogeneous of degree one on G. We consider the following Dirichlet boundary value problem of the sub-Laplace equation involving the critical exponent and singular term: $$\left\{\begin{array}{ll}-\sum_{j=1}^{m}X_j^2u(x)-\frac{a}{\|x\|^\nu}u(x)=u^{\frac{Q+2}{Q-2}}(x), x\in\Omega,\\ u(x)=0, \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,\,\, x\in \partial\Omega,\end{array}\right.$$ where ${\Omega\subset G}$ is a bounded domain with smooth boundary and ${\mathbf{0}\in\Omega}$ , Q is the homogeneous dimension of G, ${a\in \mathbb{R},\ \nu <2 }$ . We boost u to ${L^p(\Omega)}$ for any ${1\leq p < \infty}$ if ${u\in S^{1,2}_0(\Omega)}$ is a weak solution of the problem above.     

Strongly nonlinear multivalued elliptic equations on a bounded domain

In this work we study the existence of nontrivial solution for the following class of multivalued quasilinear problems $$\begin{aligned} \displaystyle -\text{ div } ( \phi (|\nabla u|) \nabla u) - b(u)u \in \lambda \partial F(x,u)\;\text{ in }\;\Omega , \quad u=0\; \text{ on }\;\partial \Omega \end{aligned}$$ where $\Omega \subset \mathbb{R }^N$ is a bounded domain, $N\ge 2$ and $\partial F(x,u)$ is a generalized gradient of $F(x,t)$ with respect to $t$ . The main tools utilized are Variational Methods for Locally Lipschitz Functional and a Concentration Compactness Theorem for Orlicz space.     

Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential

In this paper, we will prove the existence of infinitely many solutions for the following elliptic problem with critical Sobolev growth and a Hardy potential: $$-\Delta u-\frac{\mu}{|x|^2}u = |u|^{2^{\ast}-2}u+a u\quad {\rm in}\;\Omega,\quad u=0 \quad {\rm on}\; \partial\Omega,\qquad (*)$$ under the assumptions that N ≥ 7, ${\mu\in \left[0,\frac{(N-2)^2}4-4\right)}$ and a > 0, where ${2^{\ast}=\frac{2N}{N-2}}$ , and Ω is an open bounded domain in ${\mathbb{R}^N}$ which contains the origin. To achieve this goal, we consider the following perturbed problem of (*), which is of subcritical growth, $$-\Delta u-\frac{\mu}{|x|^2}u = |u|^{2^{\ast}-2-\varepsilon_n}u+au \quad {\rm in}\,\Omega, \quad u=0 \quad {\rm on}\;\partial\Omega,\qquad(\ast\ast)_n$$ where ${\varepsilon_{n} > 0}$ is small and ${\varepsilon_n \to 0}$ as n → + ∞. By the critical point theory for the even functionals, for each fixed ${\varepsilon_{n} > 0}$ small, (**) _n has a sequence of solutions ${u_{k,\varepsilon_{n}} \in H^{1}_{0}(\Omega)}$ . We obtain the existence of infinitely many solutions for (*) by showing that as n → ∞, ${u_{k,\varepsilon_{n}}}$ converges strongly in ${H^{1}_{0}(\Omega)}$ to u _k , which must be a solution of (*). Such a convergence is obtained by applying a local Pohozaev identity to exclude the possibility of the concentration of ${\{u_{k,\varepsilon_n}\}}$ .     

p-Laplacian problems with nonlinearities interacting with the spectrum

The aim of this paper is investigating the existence and the multiplicity of weak solutions of the quasilinear elliptic problem $$\left\{\begin{array}{ll}-\Delta_p u\ =\ g(x, u) \quad {\rm in} \quad \Omega,\\ u=0 \qquad \qquad \qquad {\rm on}\quad \partial\Omega,\end{array}\right.$$ where ${1 < p < + \infty, \Delta_p u = {\rm div}(|\nabla {u}|^{p-2}\nabla {u})}$ , Ω is an open bounded domain of ${\mathbb{R}^N (N \geq 3)}$ with smooth boundary ?Ω and the nonlinearity g behaves as u ^p?1 at infinity. The main tools of the proof are some abstract critical point theorems in Bartolo et al. (Nonlinear Anal. 7: 981–1012, 1983), but extended to Banach spaces, and two sequences of quasi–eigenvalues for the p–Laplacian operator as in Candela and Palmieri (Calc. Var. 34: 495–530, 2009), Li and Zhou (J. Lond. Math. Soc. 65: 123–138, 2002).     

Principal Eigenvalue of p-Laplacian Operator in Exterior Domain

We consider an eigenvalue problem of the form $$\left.\begin{array}{cl}-\Delta_{p} u = \lambda\, K(x)|u|^{p-2}u \quad \mbox{in}\quad \Omega^e\\ u(x) =0 \quad \mbox{for}\quad \partial \Omega\\ u(x) \to 0 \quad \mbox{as}\quad |x| \to \infty,\end{array} \right \}$$ where \({\Omega \subset \mathrm{I\!R\!}^N}\) is a simply connected bounded domain, containing the origin, with C ² boundary \({\partial \Omega}\) and \({\Omega^e:=\mathrm{I\!R\!^N} \setminus \overline{\Omega}}\) is the exterior domain, \({1 < p < N, \Delta_{p}u:={\rm div}(|\nabla u|^{p-2} \nabla u)}\) is the p-Laplacian operator and \({K \in L^{\infty}(\Omega^e) \cap L^{N/p}(\Omega^e)}\) is a positive function. Existence and properties of principal eigenvalue λ ₁ and its corresponding eigenfunction are established which are generally known in bounded domain or in \({\mathrm{I\!R\!}^N}\) . We also establish the decay rate of positive eigenfunction as \({|x| \to \infty}\) as well as near ?Ω.     

Uniqueness of integrable solutions to {\nabla \zeta=G \zeta, \zeta|_\Gamma = 0} for integrable tensor coefficients G and applications to elasticity

Let ${\Omega \subset \mathbb{R}^{N}}$ be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary ${\partial\Omega}$ . We show that the solution to the linear first-order system $$\nabla \zeta = G\zeta, \, \, \zeta|_\Gamma = 0 \quad \quad \quad (1)$$ is unique if ${G \in \textsf{L}^{1}(\Omega; \mathbb{R}^{(N \times N) \times N})}$ and ${\zeta \in \textsf{W}^{1,1}(\Omega; \mathbb{R}^{N})}$ . As a consequence, we prove $$||| \cdot ||| : \textsf{C}_{o}^{\infty}(\Omega, \Gamma; \mathbb{R}^{3}) \rightarrow [0, \infty), \, \, u \mapsto \parallel {\rm sym}(\nabla uP^{-1})\parallel_{\textsf{L}^{2}(\Omega)}$$ to be a norm for ${P \in \textsf{L}^{\infty}(\Omega; \mathbb{R}^{3 \times 3})}$ with Curl ${P \in \textsf{L}^{p}(\Omega; \mathbb{R}^{3 \times 3})}$ , Curl ${P^{-1} \in \textsf{L}^{q}(\Omega; \mathbb{R}^{3 \times 3})}$ for some p, q > 1 with 1/p + 1/q = 1 as well as det ${P \geq c^+ > 0}$ . We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let ${\Phi \in \textsf{H}^{1}(\Omega; \mathbb{R}^{3})}$ satisfy sym ${(\nabla\Phi^\top\nabla\Psi) = 0}$ for some ${\Psi \in \textsf{W}^{1,\infty}(\Omega; \mathbb{R}^{3}) \cap \textsf{H}^{2}(\Omega; \mathbb{R}^{3})}$ with det ${\nabla\Psi \geq c^+ > 0}$ . Then, there exist a constant translation vector ${a \in \mathbb{R}^{3}}$ and a constant skew-symmetric matrix ${A \in \mathfrak{so}(3)}$ , such that ${\Phi = A\Psi + a}$ .     

A weighted Hardy inequality and nonexistence of positive solutions

In this article, we prove that the following weighted Hardy inequality $$\begin{array}{ll}\big(\frac{|{d-p}|}{p}\big)^{p}\, \int\limits_{\Omega}\, \frac{|{u}|^{p}}{|{x}|^{p}}\;d\mu \\ \quad \quad \le \int\limits_{\Omega}\,|{\nabla u}|^{p}\;d\mu+ \big(\frac{|{d-p}|}{p}\big)^{p-1}\,\textrm{sgn}(d-p)\,\int\limits_{\Omega}|{u}|^{p}\,\frac{(x^{t}Ax)^{p/2}}{|{x}|^{p}}\; d\mu \quad \quad \quad (1) \end{array}$$ holds with optimal Hardy constant ${\big(\frac{|d-p|}{p}\big)^{p}}$ for all ${u \in W^{1,p}_{\mu,0}(\Omega)}$ if the dimension d ≥ 2, 1 < p < d, and for all ${u \in W^{1,p}_{\mu,0}(\Omega{\setminus}\{0\})}$ if p > d ≥ 1. Here we assume that Ω is an open subset of ${\mathbb{R}^{d}}$ with ${0 \in \Omega}$ , A is a real d × d-symmetric positive definite matrix, c > 0, and $$ d \mu: = \rho(x) \,dx \qquad \textrm{with} \quad \rho(x) = c \cdot \exp(-\frac{1}{p}(x^{t}Ax)^{p/2}), \quad x \in\Omega.\quad \quad (2) $$ If p > d ≥ 1, then we can deduce from (1) a weighted Poincaré inequality on ${W^{1,p}_{\mu,0}(\Omega \setminus\{0\})}$ . Due to the optimality of the Hardy constant in (1), we can establish the nonexistence (locally in time) of positive weak solutions of a p-Kolmogorov parabolic equation perturbed by a singular potential in dimension d = 1, for 1 < p <  + ∞, and when Ω =  ]0, + ∞[.     

Topological degree for solutions of fourth order mean field equations

We consider the following fourth order mean field equation with Navier boundary condition $$\Delta^2 u = \rho \frac{h(x) e^{u}}{\int_\Omega h e^{u}}\,\,{\rm in}\, \Omega,{\quad}u = \Delta u = 0\,\,{\rm on}\,\partial \Omega,\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(*)$$ where h is a C ^2,?? positive function, ?? is a bounded and smooth domain in ${\mathbb{R}^4}$ . We prove that for ${\rho \in (32m\sigma_3, 32(m + 1)\sigma_3)}$ the degree-counting formula for (*) is given by $$d(\rho)=\left\{\begin{array}{ll}\frac{1}{m!} (-\chi (\Omega) +1) \cdot\cdot \cdot (-\chi(\Omega)+m) & {\rm for}\, m >0 ,\\ 1 & {\rm for}\, m=0\end{array}\right.$$ where ??(??) is the Euler characteristic of ??. Similar result is also proved for the corresponding Dirichlet problem $$\Delta^2 u = \rho \frac{h(x) e^{u}}{\int_\Omega h e^{u}}\quad{\rm in}\,\Omega, \quad u = \nabla u = 0 \quad {\rm on}\,\,\partial \Omega.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(**)$$      

On functions whose symmetric part of gradient agree and a generalization of Reshetnyak’s compactness theorem

We consider the following question: Given a connected open domain ${\Omega \subset \mathbb{R}^n}$ , suppose ${u, v : \Omega \rightarrow \mathbb{R}^n}$ with det ${(\nabla u) > 0}$ , det ${(\nabla v) > 0}$ a.e. are such that ${\nabla u^T(x)\nabla u(x) = \nabla v(x)^T \nabla v(x)}$ a.e. , does this imply a global relation of the form ${\nabla v(x) = R\nabla u(x)}$ a.e. in Ω where ${R \in SO(n)}$ ? If u, v are C ¹ it is an exercise to see this true, if ${u, v\in W^{1,1}}$ we show this is false. In Theorem 1 we prove this question has a positive answer if ${v \in W^{1,1}}$ and ${u \in W^{1,n}}$ is a mapping of L ^p integrable dilatation for p > n ? 1. These conditions are sharp in two dimensions and this result represents a generalization of the corollary to Liouville’s theorem that states that the differential inclusion ${\nabla u \in SO(n)}$ can only be satisfied by an affine mapping. Liouville’s corollary for rotations has been generalized by Reshetnyak who proved convergence of gradients to a fixed rotation for any weakly converging sequence ${v_k \in W^{1,1}}$ for which $$\int \limits_{\Omega} {\rm dist}(\nabla v_k, SO(n))dz \rightarrow 0 \, {\rm as} \, k \rightarrow \infty.$$ Let S(·) denote the (multiplicative) symmetric part of a matrix. In Theorem 3 we prove an analogous result to Theorem 1 for any pair of weakly converging sequences ${v_k \in W^{1,p}}$ and ${u_k \in W^{1,\frac{p(n-1)}{p-1}}}$ (where ${p \in [1, n]}$ and the sequence (u _k ) has its dilatation pointwise bounded above by an L ^r integrable function, r > n ? 1) that satisfy ${\int_{\Omega} |S(\nabla u_k) - S(\nabla v_k)|^p dz \rightarrow 0}$ as k → ∞ and for which the sign of the det ${(\nabla v_k)}$ tends to 1 in L ¹. This result contains Reshetnyak’s theorem as the special case (u _k ) ≡ Id, p = 1.     

Multiple positive solutions of singular and critical elliptic problem in {\mathbb{R}^2} with discontinuous nonlinearities

Let Ω be a bounded domain in ${\mathbb{R}^2}$ with smooth boundary. We consider the following singular and critical elliptic problem with discontinuous nonlinearity: $$(P_\lambda)\left \{\begin{array}{ll} - \Delta u = \lambda \left(\frac{m(x, u) e^{\alpha{u}^2}}{|x|^{\beta}} + u^{q}g(u - a)\right),\quad{u} > 0 \quad {\rm in} \quad \Omega\\u \quad \quad = 0\quad {\rm on} \quad \partial \Omega \end{array}\right.$$ where ${0\leq q < 1 ,0< \alpha\leq4\pi}$ and ${\beta \in [0, 2)}$ such that ${\frac{\beta}{2} + \frac{\alpha}{4\pi} \leq 1}$ and ${{g(t - a) = \left\{\begin{array}{ll}1, t \leq a\\ 0, t > a.\end{array}\right.}}$ Under the suitable assumptions on m(x, t) we show the existence and multiplicity of solutions for maximal interval for λ.     

Stability estimates for nonlinear hyperbolic problems with nonlinear Wentzell boundary conditions

Of concern is the nonlinear hyperbolic problem with nonlinear dynamic boundary conditions $$\left\{ \begin{array}{lll} u_{tt} ={\rm div} (\mathcal{A} \nabla u)-\gamma (x,u_t), && \quad {\rm in} \; (0, \infty) \times \Omega,\\ u(0, \cdot)=f, \, u_t(0,\cdot)=g, && \quad {\rm in}\; \Omega, \\ u_{tt} + \beta \partial^ \mathcal{A}_\nu u+c(x)u+ \delta (x,u_t)-q \beta \Lambda_{\rm LB} u=0,&& \quad {\rm on} \;(0, \infty ) \times \partial \Omega . \end{array}\right. $$ for t ≥  0 and ${x \in \Omega \subset \mathbb{R}^N}$ ; the last equation holds on the boundary ?Ω. Here ${\mathcal{A}= \{a_{ij}(x)\}_{ij}}$ is a real, hermitian, uniformly positive definite N × N matrix; ${\beta \in C(\partial \Omega)}$ , with β > 0; ${\gamma:\Omega \times \mathbb{R} \to \mathbb{R}; \delta:\partial \Omega \times \mathbb{R} \to \mathbb{R}; \,c:\partial \Omega \to \mathbb{R}; \, q \ge 0, \Lambda_{\rm LB}}$ is the Laplace–Beltrami operator on ?Ω, and ${\partial^\mathcal{A}_\nu u}$ is the conormal derivative of u with respect to ${\mathcal{A}}$ ; everything is sufficiently regular. We prove explicit stability estimates of the solution u with respect to the coefficients ${\mathcal{A},\,\beta,\,\gamma,\,\delta,\,c,\,q}$ , and the initial conditions f, g. Our arguments cover the singular case of a problem with q = 0 which is approximated by problems with positive q.     

Radial and non radial solutions for Hardy–Hénon type elliptic systems

We discuss existence and non-existence of positive solutions for the following system of Hardy and Hénon type: $$\left\{\begin{array}{ll} {-\Delta v=|x|^{\alpha}u^{p},\,-\Delta u=|x|^{\beta}v^{q} \,\,{\rm in}\, \Omega,}\\ {u=v=0 \quad\quad\quad\quad\quad\quad\quad\quad\quad{\rm on}\, \partial \Omega}, \end{array}\right.$$ where ${\Omega\ni 0}$ is a bounded domain in ${\mathbb{R}^{N}}$ , N ≥ 3, p, q > 1, and α, β > ?N. We also study symmetry breaking for ground states when Ω is the unit ball in ${\mathbb{R}^{N}}$ .     

Existence and multiplicity results for Pucci’s operators involving nonlinearities with zeros

We study the problem $$ \left\{\begin{array}{ll} {-\varepsilon^{2}\mathcal{M}^+_{\lambda,\Lambda}(D^{2}u) = f (x, u)} \quad\; {\rm in} \; \Omega,\\ {u = 0} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad {\rm on} \; \partial{\Omega}, \end{array} \right.$$ where Ω is a smooth bounded domain in ${\mathbb{R}^{N},N > 2,}$ and show it possesses nontrivial solutions for small values of ε provided f is a nonnegative continuous function which has a positive zero. The multiplicity result is based on degree theory together with a new Liouville type theorem for ${-{M}^+_{\lambda,\Lambda}(D^{2}u) = f(u)}$ in ${\mathbb{R}^{N}}$ for nonnegative nonlinearities with zeros.     

Quasilinear elliptic equations in \mathbb{R }^{N} via variational methods and Orlicz-Sobolev embeddings

In this paper we prove the existence of a nontrivial non-negative radial solution for the quasilinear elliptic problem $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\nabla \cdot \left[\phi ^{\prime }(|\nabla u|^2)\nabla u \right] +|u|^{\alpha -2}u =|u|^{s-2} u,&x\in \mathbb{R }^{N},\\ u(x) \rightarrow 0, \quad \text{ as} |x|\rightarrow \infty , \end{array} \right. \end{aligned}$$ where $N\ge 2, \phi (t)$ behaves like $t^{q/2}$ for small $t$ and $t^{p/2}$ for large $t, 1< p<q<N, 1<\alpha \le p^* q^{\prime }/p^{\prime }$ and $\max \{q,\alpha \}< s<p^*,$ being $p^*=\frac{pN}{N-p}$ and $p^{\prime }$ and $q^{\prime }$ the conjugate exponents, respectively, of $p$ and $q$ . Our aim is to approach the problem variationally by using the tools of critical points theory in an Orlicz-Sobolev space. A multiplicity result is also given.     

Perturbed asymptotically linear problems

The aim of this paper is to investigate the existence of solutions of the semilinear elliptic problem $$\left\{\begin{array}{ll} -\Delta u\ =\ p(x, u) + \varepsilon g(x, u)\quad {\rm in}\,\, \Omega, \\ u=0 \quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,\,\,\,\,{\rm on}\,\, \partial\Omega, \end{array} \right. \quad\quad\quad(0.1) $$ where Ω is an open bounded domain of ${\mathbb{R}^N}$ , ${\varepsilon\in\mathbb{R}, p}$ is subcritical and asymptotically linear at infinity, and g is just a continuous function. Even when this problem has not a variational structure on ${H^1_0(\Omega)}$ , suitable procedures and estimates allow us to prove that the number of distinct critical levels of the functional associated to the unperturbed problem is “stable” under small perturbations, in particular obtaining multiplicity results if p is odd, both in the non-resonant and in the resonant case.     

Large viscosity solutions for some fully nonlinear equations

We study existence, uniqueness and asymptotic behavior near the boundary of solutions of the problem $$\left\{\begin{array}{ll}-F(D^{2} u) + \beta (u) = f \quad {\rm in} \, \Omega, \\ u = + \infty \quad \quad \quad \quad \quad \quad \,\,\,\, {\rm on}\, \partial \Omega, \end{array} \right.\quad \quad \quad \quad \quad {\rm (P)}$$ where Ω is a bounded smooth domain in ${{\mathbb R}^N, N >1 , F}$ is a fully nonlinear elliptic operator and β is a nondecreasing continuous function. Assuming that β satisfies the Keller–Osserman condition, we obtain existence results which apply to ${f \in L^\infty_{loc}(\Omega)}$ or f having only local integrability properties where viscosity solutions are well defined, i.e. ${f \in L^N_{loc}(\Omega)}$ . Besides, we find the asymptotic behavior near the boundary of solutions of (P) for a wide class of functions ${f \in \mathcal{C}(\Omega)}$ . Based in this behavior, we also prove uniqueness.